Dissipation and fluctuation for a randomly kicked particle: Normal and anomalous diffusion

A stochastic model is considered which describes the motion of a one-dimensional classical particle kicked at random by heat bath particles. The effect of the distribution of the time intervals between the collision events on the motion of the test particle is studied. It is shown that the general pattern strongly depends on the long time behavior of the waiting time distribution function. The Brownian or normal diffusive type of motion is obtained when the distribution function has finite moments. If, however, all these moments diverge the diffusion has an anomalous character. The relaxation is characterized by power law dependences, while the mean square displacement is dominated by the ballistic contribution. The second, superdiffusive term can be related to the momentum relaxation by a relation differing from the conventional Einstein relation. It is also important to emphasize that the information on the initial state of the test particle is essential for the long time behavior of the mean square displacement for the proper formulation of the dissipation-fluctuation relation.